What is the integral from 0 to 1 of xsin(pix) ?

The integral from 0 to 1 of xsin(pix) is 1/pi

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integral from 0 to 1 of xsin(pix)

\int_{0}^{1} x sin (π x) d x

\frac{1}{π}

Decimal

0.31830 \dots

Solution steps

\int_{0}^{1} x sin (π x) d x

Apply Integration By Parts

= [- \frac{1}{π} x cos (π x) - \int - \frac{1}{π} cos (π x) d x]_{0}^{1}

\int - \frac{1}{π} cos (π x) d x = - \frac{1}{π ^{2}} sin (π x)

= [- \frac{1}{π} x cos (π x) - (- \frac{1}{π ^{2}} sin (π x))]_{0}^{1}

Simplify

= [- \frac{1}{π} x cos (π x) + \frac{1}{π ^{2}} sin (π x)]_{0}^{1}

Compute the boundaries:

\frac{1}{π}

= \frac{1}{π}

What is the integral from 0 to 1 of xsin(pix) ?
The integral from 0 to 1 of xsin(pix) is 1/pi